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In this study, we proposed a family of m-point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling.

Abstract Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler’s fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution’s existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example.

In this paper, we investigate the existence of a solution for the fractional q-integro-differential inclusion with new double sum and product boundary conditions. One of the most recent techniques of fixed point theory, i.e. endpoints property, and inequalities, plays a central role in proving the main results. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures. The paper ends with an example.

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.

In this paper, an analysis of a mathematical model of the coronavirus is carried out by using two fractal-fractional parameters. This dangerous virus infects a person through the mouth, eyes, nose or hands. This makes it so dangerous that no one can get rid of it. One of the main factors contributing to increasing infections of this deadly virus is crowding. We believe that it is necessary to model this effect mathematically to predict the possible outcomes. Hence, the study of neural network-based models related to the spread of this virus can yield new results. This paper also introduces the use of artificial neural networks (ANNs) to approximate the solutions, which is a significant contribution in this regard. We suggest employing this new method to solve a system of integral equations that explain the dynamics of infectious diseases instead of the classical numerical methods. Our study shows that, compared to the Adams-Bashforth algorithm, the ANN is a reliable candidate for solving the problems.

We aim to investigate an integro-differential inclusion using a novel computational approach in this research. The use of quantum calculus, and consequently the creation of discrete space, allows the computer and computational algorithms to solve our desired problem. Furthermore, to guarantee the existence of the solution, we use the endpoint property based on fixed point methods, which is one of the most recent techniques in fixed point theory. The above will show the novelty of our work, because most researchers use classical fixed point techniques in continuous space. Moreover, the sensitivity of the parameters involved in controlling the existence of the solution can be recognized from the heatmaps. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures in our examples that are presented at the end of the work.

A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new $ \psi $-Caputo operator. The novelty of this work, in addition to the $ \psi $-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the $ \psi(\kappa) $ function.

In this paper, we investigate a multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains $ (a, \infty), a\geq 0 $. The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers ($ \mathcal UH $), Ulam-Hyers-Rassias ($ \mathcal UHR $) and semi-Ulam-Hyers-Rassias (s-$ \mathcal UHR $) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results.

The present study aims to consider a mathematical eco-epidemiological model involving two fractional operators. To this end, we provide approximate solutions to these fractional systems through the application of a numerical technique that is based on the rule of product integration. This feature contributes greatly to the efficiency and effectiveness of both methods. We have also presented some theoretical discussions related to the equilibrium points of the system. Further, several numerical simulations are presented in order to illustrate the impact of choosing different parameters on the dynamics of the model. It is demonstrated that the obtained numerical results are completely consistent with the expected theoretical results. Moreover, both techniques can be used to solve other problems in epidemiology and describe other problems in the future. The article's model has never been studied via the employed fractional operators, and this is a distinct point for our work and other existing research.

Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective.

Abstract In this manuscript, we study the existence and uniqueness of solutions for a new neutral hybrid nonlinear differential equation in the context of a fractional generalized operator in the sense of ψ-Caputo. To emphasize the novelty of the manuscript, a pure technique of the noncompactness measures is applied to a hybrid system based on the notion of the modulus of continuity in Darbo’s criterion that covers the existing results of other works published before. The Ulam–Hyers and generalized Ulam–Hyers stabilities are explored for the given neutral nonhybrid nonlinear problem. An application is prepared in the framework of an example to ensure the validity of theorems for different cases.

In computational mathematics, the comparison of convergence rate in different iterative methods is an important concept from theoretical point of view. The importance of this comparison is relevant for researchers who want to discover which one of these iterations converges to the fixed point more rapidly. In this article, we study the different numerical methods to calculate fixed point in digital metric spaces, introduce a new k-step iterative process and conduct an analysis on the strong convergence, stability and data dependence of the mentioned scheme. Some illustrative examples are given to show that this iteration process converges faster.

We prove existence and uniqueness of solutions to discrete fractional equations that involve Riemann-Liouville and Caputo fractional derivatives with three-point boundary conditions. The results are obtained by conducting an analysis via the Banach principle and the Brouwer fixed point criterion. Moreover, we prove stability, including Hyers-Ulam and Hyers-Ulam-Rassias type results. Finally, some numerical models are provided to illustrate and validate the theoretical results.

In this paper, for the first time, we study the existence and uniqueness of solutions of a Caputo variable-order initial value problem (IVP) in the impulsive settings. Our existence results are proved by using two fixed point theorems. The Ulam-Hyers stability of solutions is established for the variable order impulsive initial value problem. Finally, we provide an example to show the correctness of the results.

This work is concerned with the study of the existing solution for the fractional (p,q)-difference equation under first order (p,q)-difference boundary conditions in generalized metric space. To achieve the solution, we combine some contraction techniques in fixed point theory with the numerical techniques of the Lipschitz matrix and vector norms. To do this, we first associate a matrix to a desired boundary value problem. Then we present sufficient conditions for the convergence of this matrix to zero. Also, we design some algorithms to use the computer for calculate the eigenvalues of such matrices and different values of (p,q)-Gamma function. Finally, by presenting two numerical examples, we examine the performance and correctness of the proposed method. Some tables and figures are provided to better understand the issues.

Malaria disease, which is of parasitic origin, has always been one of the challenges for human societies in areas with poor sanitation. The lack of proper distribution of drugs and lack of awareness of people in such environments cause us to see many deaths every year, especially in children under the age of five. Due to the importance of this issue, in this paper, a new five-compartmental $ (c_1, c_2) $-fractal-fractional $ \mathcal{SIR} $-$ \mathcal{SI} $-model of malaria disease for humans and mosquitoes is presented. We use the generalized Mittag-Leffler fractal-fractional derivatives to design such a mathematical model. In different ways, we study all theoretical aspects of solutions such as the existence, uniqueness and stability. A Newton polynomial that works in fractal-fractional settings is shown, which allows us to get some numerical trajectories. From the trajectories, we saw that an increase in antimalarial treatment in consideration to memory effects reduces the peak of sick individuals, and mosquito insecticide spraying minimizes the disease burden in all compartments.

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.

This paper reports a modified F-iterative process for finding the fixed points of three generalized α-nonexpansive mappings. We assume certain assumptions to establish the weak and strong convergence of the scheme in the context of a Banach space. We suggest a numerical example of generalized α-nonexpansive mappings which exceeds, properly, the category of functions furnished with a condition (C). After that, we show that our modified F-iterative scheme of this example converges to a common fixed point of three generalized α-nonexpansive mappings. As an application of our main findings, we suggest a new projection-type iterative scheme to solve variational inequality problems in the setting of generalized α-nonexpansive mappings. The main finding of the paper is new and extends many known results of the literature.

One of the most famous equations that are widely used in various branches of physics, mathematics, financial markets, etc. is the Langevin equation. In this work, we investigate the existence of the solution for two generalized fractional hybrid Langevin equations under different boundary conditions. For this purpose, the problem of the existence of a solution will become the problem of finding a fixed point for an operator defined in the Banach space. To achieve the result, one of the recent fixed point techniques, namely the α-ψ-contraction technique, will be used. We provide sufficient conditions to use this type of contraction in our main theorems. In the calculations of the auxiliary lemmas that we present, the Mittag-Leffler function plays a fundamental role. The fractional derivative operators used are of the Caputo type. Two examples are provided to demonstrate the validity of the obtained theorems. Also, some figures and a table are presented to illustrate the results.

The goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it apply the Aboodh transform along with the residual power series method (RPSM). It is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates the computation of the fractional derivatives each time. As ARPSM just requires the idea of an infinite limit, we simply need a few computations to get the coefficients. This technique solves nonlinear problems without the He's polynomials and Adomian polynomials, so the small size of computation of this technique is the strength of the scheme, which is an advantage over the homotopy perturbation method and the Adomian decomposition method. The absolute and relative errors of five linear and non-linear problems are numerically examined to determine the efficacy and accuracy of ARPSM for time-fractional PDEs with variable coefficients. In addition, numerical results are also compared with other methods such as the RPSM and the natural transform decomposition method (NTDM). Some graphs are also plotted for various values of fractional orders. The results show that our technique is easy to use, accurate, and effective. Mathematica software is used to calculate the numerical and symbolic quantities in the paper.

Abstract In this paper, we study a new class of non-hybrid single-valued fractional boundary value problems equipped with integro-non-hybrid-multiterm-multipoint-multistrip conditions and a fully hybrid integro-multi-valued fractional boundary value problem by some new methods including the Kuratowski measures based on Sadovskii’s theorem, Krasnoselskii–Zabreiko criterion, and Dhage’s technique. We generalize the Gronwall inequality in relation to a non-hybrid single-valued fractional boundary value problem, and then we investigate the stability notions in two versions. To examine the correctness of the results, we provide some examples.

This paper discusses the existence, uniqueness and stability of solutions for a nonlinear fractional differential system consisting of a nonlinear Caputo-Hadamard fractional initial value problem (FIVP). By using some properties of the modified Laplace transform, we derive an equivalent Hadamard integral equation with respect to one-parametric and two-parametric Mittag-Leffer functions. The Banach contraction principle is used to give the existence of the corresponding solution and its uniqueness. Then, based on a Lyapunov-like function and a K-class function, the generalized Mittag-Leffler stability is discussed to solve a nonlinear Caputo-Hadamard FIVP. The findings are validated by giving an example.

Abstract Results reported in this paper establish the existence of solutions for a class of generalized fractional inclusions based on the Caputo–Hadamard jerk system. Under some inequalities between multi-functions and with the help of special contractions and admissible maps, we investigate the existence criteria. Fixed points and end points are key roles in this manuscript, and the approximate property for end points helps us to derive the desired result for existence theory. An example is prepared to demonstrate the consistency and correctness of analytical findings.

Abstract A thermostat model described by a second-order fractional difference equation is proposed in this paper with one sensor and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality. By means of well-known contraction mapping and the Brouwer fixed-point theorem, we provide new results on the existence and uniqueness of solutions. In this work by use of the Caputo fractional difference operator and Hyer–Ulam stability definitions we check the sufficient conditions and solution of the equations to be stable, while most researchers have examined the necessary conditions in different ways. Further, we also establish some results regarding Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability for our discrete fractional-order thermostat models. To support the theoretical results, we present suitable examples describing the thermostat models that are illustrated by graphical representation.

In this article, we consider a fractal-fractional tobacco mathematical model with generalized kernels of Mittag-Leffler functions for qualitative and numerical studies. From qualitative point of view, our study includes; existence criteria, uniqueness of solution and Hyers-Ulam stability. For the numerical aspect, we utilize Lagrange's interpolation polynomial and obtain a numerical scheme which is further illustrated simulations. Lastly, a comparative analysis is presented for different fractal and fractional orders. The numerical results are divided into four figures based on different fractal and fractional orders. We have found that the fractional and fractal orders have a significant impact on the dynamical behaviour of the model.

In this paper, we study a general system of fractional hybrid differential equations with a nonlinear $ \phi_p $-operator, and prove the existence of solution, uniqueness of solution and Hyers-Ulam stability. We use the Caputo fractional derivative in this system so that our system is more general and complex than other nonlinear systems studied before. To establish the results, Green functions are used to transform the considered hybrid boundary problem into a system of fractional integral equations. Then, with the help of the topological degree theorem, we derive some sufficient conditions that ensure the existence and uniqueness of solutions for the proposed system. Finally, an example is presented to show the validity and correctness of the obtained results.

Abstract The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring to fractional theories, resolvent operators, semigroup theory, Gronwall’s inequality, and Lipschitz condition. The article avoids the use of well-known fixed point theorem approaches. We have also included one example of theoretical consequences that has been validated.

Abstract In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Abstract The research presents a qualitative investigation of a fractional-order consumer-resource system with the hunting cooperation interaction functional and an infection developed in the resources population. The existence of the equilibria is discussed where there are many scenarios that have been distinguished as the extinction of both populations, the extinction of the infection, the persistence of the infection, and the two populations. The influence of the hunting cooperation interaction functional is also investigated where it can influence the existence of equilibria and their stability. A proper numerical scheme is used for building a proper graphical representation for the goal of confirming the theoretical results.

Abstract In this manuscript, we deal with the nonlocal controllability results for the fractional evolution system of 1 < r < 2 $1< r

Abstract In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example.

Abstract This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the extended fractional snap problem by means of the G $\mathbb{G}$ -operators. After finding its equivalent solution as a form of the integral equation, we establish the existence criterion of this reformulated model with respect to some known fixed point techniques. Then we analyze its stability and further investigate the inclusion version of the problem with the help of some special contractions. We present numerical simulations for solutions of several examples regarding the fractional G $\mathbb{G}$ -snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders.

This paper derives some equalities via twice differentiable functions and conformable fractional integrals. With the help of the obtained identities, we present new trapezoid-type and midpoint-type inequalities via convex functions in the context of the conformable fractional integrals. New inequalities are obtained by taking advantage of the convexity property, power mean inequality, and Hölder’s inequality. We show that this new family of inequalities generalizes some previous research studies by special choices. Furthermore, new other relevant results with trapezoid-type and midpoint-type inequalities are obtained.

Abstract In this paper, we introduce a new structure of the generalized multi-point thermostat control model motivated by its standard model. By presenting integral solution of this boundary problem, the existence property along with the uniqueness property are investigated by means of a special version of contractions named μ-φ-contractions and the Banach contraction principle. Then, on the given nonlinear generalized BVP of thermostat, the Bernstein polynomials are introduced and numerical solutions obtained by them are presented. At the end, three different structures of nonlinear thermostat models are designed and the results are examined.

Abstract The goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fractional theory to obtain the main results by Monch’s fixed point technique. Moreover, we show the existence and controllability of the thought about the fractional system with the nonlocal condition. In addition, an application to illustrate the outcomes is also included.

Abstract In this work, we study a q-differential inclusion with doubled integral boundary conditions under the Caputo derivative. To achieve the desired result, we use the endpoint property introduced by Amini-Harandi and quantum calculus. Integral boundary conditions were considered on time scale T t 0 = { t 0 , t 0 q , t 0 q 2 , … } ∪ { 0 } $\mathcal{T}_{t_{0}}=\{t_{0},t_{0}q,t_{0}q^{2}, \ldots\}\cup \{0\}$ . To better evaluate the validity of our results, we provided an example, some graphs, and tables.

Abstract In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

Abstract Some complicated events can be modeled by systems of differential equations. On the other hand, inclusion systems can describe complex phenomena having some shocks better than the system of differential equations. Also, one of the interests of researchers in this field is an investigation of hybrid systems. In this paper, we study the existence of solutions for hybrid and non-hybrid k-dimensional sequential inclusion systems by considering some integral boundary conditions. In this way, we use different methods such as α-ψ contractions and the endpoint technique. Finally, we present two examples to illustrate our main results.

Abstract The fundamental goal of the study under consideration is to establish some of the existence criteria needed for a particular fractional inclusion model of cantilever beam in the setting of quantum calculus using new arguments of existence theory. In this way, we investigate a fractional integral equation that corresponds to the aforementioned boundary value problem. In a more concrete sense, we design new multi-valued operators based on this integral equation, which belong to the certain subclasses of functions, called α-admissible and α-ψ-contractive multi-functions, in combination with the AEP-property. Also, we use some inequalities such as Ω-inequality and set-valued version inequalities. Moreover, we add a simulative example for a numerical analysis of our results obtained in this study.

Abstract A memristor is naturally a nonlinear and at the same time memory element that may substitute resistors for next-generation nonlinear computational circuits that can show complex behaviors including chaos. A four-dimensional memristor system with the Atangana–Baleanu fractional nonsingular operator in the sense of Caputo is investigated. The Banach fixed point theorem for contraction principle is used to verify the existence–uniqueness of the fractional representation of the given system. A newly developed numerical scheme for fractional-order systems introduced by Toufik and Atangana is utilized to obtain the phase portraits of the suggested system for different fractional derivative orders and different parameter values of the system. Analysis on the local stability of the fractional model via the Matignon criteria showed that the trivial equilibrium point is unstable. The dynamics of the system are investigated using Lyapunov exponents for the characterization of the nature of the chaos and to verify the dissipativity of the system. It is shown that the supposed system is chaotic and it is significantly sensitive to parameter variation and small initial condition changes.

Abstract In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455–7462, 2021) for a family of second-order semilinear control system by formulating a sequence of piecewise controls. This approach does not involve large estimations which are required to apply fixed point theorems. Therefore, we avoid the use of fixed point theory and the contraction mapping principle. We establish that a second-order semilinear system drives any starting position to the required final position from the domain of the system. To achieve the required results, we suppose that the linear system is exactly controllable at every non-initial time period, the norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function, and the nonlinear term is bounded. Finally, an example has been presented to validate the results.

Abstract In this paper, we consider a new coupled system of fractional boundary value problems based on the thermostat control model. With the help of fixed point theory, we investigate the existence criterion of the solution to the given coupled system. This property is proved by using the Krasnoselskii’s fixed point theorem and its uniqueness is proved via the Banach principle for contractions. Further, the Hyers–Ulam stability of solutions is investigated. Then, we find the approximate solution of the coupled fractional thermostat control system by using a numerical technique called the generalized differential transform method. To show the consistency and validity of our theoretical results, we provide two illustrative examples.

Abstract The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is obtained by using the Banach principle. For obtaining two positive solutions, we apply another fixed point criterion due to Avery–Anderson–Henderson on cones by establishing some inequalities. An illustrative example is presented to indicate the validity of the obtained results. The results are new and provide a generalization to some known results in the literature.

Abstract In this paper, we study the existence of solutions for a generalized sequential Caputo-type fractional neutral differential inclusion with generalized integral conditions. The used fractional operator has the generalized kernel in the format of ( ϑ ( t ) − ϑ ( s ) ) $( \vartheta (t)-\vartheta (s)) $ along with differential operator 1 ϑ ′ ( t ) d d t $\frac{1}{\vartheta '(t)}\,\frac{\mathrm{d}}{\mathrm{d}t}$ . We obtain existence results for two cases of convex-valued and nonconvex-valued multifunctions in two separated sections. We derive our findings by means of the fixed point principles in the context of the set-valued analysis. We give two suitable examples to validate the theoretical results.

Abstract The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t. q-RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital α-admissible and α-ψ-contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.

Abstract In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.

Abstract In this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

Abstract The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

Abstract In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

Abstract By using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by utilizing Laguerre wavelets in conjunction with the Adomian decomposition method. We present the procedure of implementation and convergence analysis for the method. This method is tested on fractional Fisher’s equation and the singular fractional Emden–Fowler equation. We compare the results produced by the present method with some well-known results.